Calculating instrument



Feb. 28, 1928.

CALCULATING INSTRUMENT Filed May 1. 1926 L un@ 2 Sheets-Sheet 1 J. E. ROWE CALCULATING INSTRUMENT Feb. 28, 1928.

Filed May 1V 1926 2 Sheets-snee?. 2

.lill/lll urli 114 MAJ/mf @Hoz neig iengineering, but also @Panarea y Fes. as, 192s.

, J'osarn wenn zown, 'on WIr-museum; vindnu l cancumrruemsrnumr.

,Application 'med lay l,

M y invention relates to instruments for solvlng mathematical roblems, and for use in surveyin range nding, etc.; andthe particular oject of my invention is to provide an instrument by which plane triangles ot all descriptions may be rapidly and accurately solved, their areas and altitudes found, and arithmetical o erations depending uponv the similarity o triangles rapidly performed. f V

Itis also my object to produce an instru- `ment such that if any three determining parts of a triangle are set, the other parts of the said triangle may be at onceread, f

It is further the ob'ect of my invention not only to simplify ca culations in practical to simplify and humanize the teaching of mathematlcs.y

Some of the uses for my instrument in ractical engineerin are the automatic S0- ution of triangles 1n ordinary` surveying; finding the area of anv triangle as the pro uct of the base and half altitude; the automatic solution-of all sextant problems, such as finding the altitude and distance of a visible object; range finding both on land and sea; and in the rapid solution of trigonometrical equations and thev finding of the six trigonometric functions of any angle.

In simplifying the teaching of mathematics it enables one to combine the arts of plane geometry and trigonometry under one subject head, 'which I term trigonometry and in such a manner that any problem in geometry involving triangles may be presented by the instructor in a very attractive way; and so that the truth of the solution of a problem is at once seen by the student. With the use of my trinometer it will be possible to cover more ground in mathematics in a given time; and it will aid the student in acquiring in a shorter time Va thorough understanding -of the science.

Among the arithmetical operations which my trinomcter may be used to perform are the multiplication of two numbers, the division of one number by another; extract-l ing the square root of a number;.and, in

fact, many arithmetical processes that can be made 'dependent-upon similarity 'of tri-l angles. u l

`Referring tothe drawings, which lform a part of this specification: Figure 1 shows a plan of my trinometer; Figure vZis a detail showing in plan the head' of the instrument;

ma smaliro. 10am. Figure a is Q re 4 shows in detail one of the sli, mg mem 5 1s a crossl section of the instrument taken on the lme 5-75 in Figure 4; and Figure .6 is a4cross section taken on line 6 6 in Figure In Figure 1, E-F, is a straight bar 4having on one slde the beveled slot One of the edges of this bar for convenience the lower one, is graduated as shown Aat H. In some cases I graduate the upper edge of the bar, as described later. Siding vupon Athe bar E-F, which for the sake of brevity 1 refer to as the base-bar, is thewmechanism D; which consists of a plate I,'on the u per art of which isl mounted a rotractor an on the lower part of whic is engraved a. vernier scale K arranged to ride against the `,graduated edge of the base-bar. Mounted within and concentrically with the protractor J is a revolving tached a vernier M which lies against the scale engraved on the circular limb of the protractor J. On the revolving late L is plate L,` towhich is ata section of -thehead shown in rs of the instrument; Figure mounted the sheathjN through w ich slides a scale-arm O. The graduated side of the scale-arm O passes directl through the center of the axis of the revolving plate L; and on the said plate L there is engraved a vermer S which lies against the aduated scale-arm O. The movement of t e slide I along -'the base-bar is controlled by the screw P; the revolution of the plate L inthe rotractor is governed by the screw Q, an the sliding of the scale-arm O through the sheath N is governed by the screw R. On the sliding Vernier K the zero is located exact] under the center of the rotractor J. The zero of the Vernier S is lbcated directly on the center of the protractor J. The zero of the- Vernier M isl located on the de e graduation of the protractor J. It wil be -noted that the 90 degree graduation is to one the sheath N1 is indicated by the letter f raduationlof the protractor, the `scale-ariir` il sliding through the' sheath N, will at yall) Thisarmitwillbeseen left hand side ;which sidej-passes throughthe I center of the bar; with means for iixin iltractor l 1 1 At Agis-aft d protractor J, similarin most of'its details to the protractors at D and B; In this case the scale-arm U lies,

protractor; and4 against the rear face ofythe the zeroof the vern1erM11sarranged on the center lineof threvolvin l.plate Lso4 :V

that when Ait ycoincidesfjwith-j't e 90 degree ri ht angles to thebase-bar E-F y A. Jn Figures lg-a'nd .5 the` protractorJ is shown xed on the back of the base# ar' H is ,marked `exactly below the centerpof the'protractorr t J 2'; both z ero'and center lying line o-fL' yIn some instruments EF; and the zero of the Iscale purposes it Ais preferredto makethe `protractor -J2 slidable on the rearV of the baseit in any desired. position on the 'basear. This construction is described'and claimed in a separate application `filed May 9, 1927, Serial Number 189,882,entitled Mathematical inf .struments The centers of each ofthethree protractors J, J1 and J 2,' and the centers of each yof kthe three revolving plates L, L1, and L2 are on the'line o45, and are `at thesarue distance frein/the base-bar E-F. In -Figure 1 the base ofthe triangle CDB= lies between the` l'centers of preti-'actors J and J1-and passes protractors J, and J2,

through the center ot' protraetor J2. That is, the kbase oi the triangle connects the three centers; and it is lprojected togthe scale H or'm'easurement, f v. y e

The graduated edges of scale-arms O, U

and yTft-which pass-through the'. centers of respectively,

1 and' 1v converge towards the bead C; and'rf proi jected `would ypass directly through. the cen- 1 :ter 'of the pivot of the head C.- This head C f 'is/engraved ai scale A i this scalefVV is countersunlgas at V1, so that ther graduationsof which are engraved "of the scale-arms- Consists otra disc or protracto-r V, `on which W. vThe' space within the verniers Xand X1, on the headsY andiYu O and T,

aga-inst, rather. than overlap, -the graduatfons'on scale The central fthe usual Way on the pivot Z,

thcheads Y, and Y1 are fitted ktogether in as shown in Figures'l and ,3, so that the scale-arms O and T, and their headfverniers X and X1,

`made integral may lieinvthe same plane. The'movement vof arms OjandT aboutthe pivot Z may be lockedy by means of the screws 02,) T1;` MovementofO and yT may also be con` 'trolled by the clamping nut Z1,'Whicherry ygages the threaded endof the ypivot Z; as shown yinFigure 3.k y f' The upper end of the with the protractorV; or it is graduated on its' angles to on the dottedy for special DCA;

may abut portions of scale-arm U may be,

'neel-,ossi t 1 maylfterininate inv a head iwhich is riveted or screwed -to the protractor V, as shownin Y Figures 2 and 3. Ig do not limit the attach ment oferm U to rotractor V to the rigid form, for it may -or, certain uses be found to be more convenient to pivot the scale-arm --U onthe back of the i rotractor V, cont-rol-g respect to protractor. g screwyin the usual ling its movement wit by means ofa sett way.

The-protracter J2. .isfgraduated so that.` lthe degree division lies on aline at right f 'the base-bar E-F; andthe zero lof the Vernier Mz lies on the ysame line when thel scale-armi U is "at lrightangles tothe ybase-bar E-F. On protractors J and J,

the 90 degree divisions are to the right and left, respectively, `of the line at right angies to the base-bar, because the zeros of the verniers M and M1 must `not be hidden by the sheathsy N and N1, which areJ on the fronts or near sidesof the plates Land L1,

respectively.` 'On the protractor V the zeros of the` circular scale W are located so that the zeros on the verniers X and X1 on the heads Y and Y1 of will register with ythem when the graduated edges of the scalefarms with the graduated edgeof the scale-arm U. Consequently the reading of the Vernier X will give'the angle between scale-arms O and U; and the reading ofthe Vernier X1- .of thepivot. Z asfC,`the center of the pro-` With baser AB.

'To measure the'altitude and the areay of triangle ACB, and to determine the three angles, ther three sides having been given:

first the clamping screws Q, R, Q1, R1, Q1,l

R2, P, P1 and 02,; T2, are loosened; the scale- 'arm Ui's slidj throughr the sheath N, until the distanceA CAiS equivalent to the `length i ot'Y ka side" of the triangle, and clamping serevvv R2clamps it in placek in ythe sheath VN 2; the scale-arm T is 'slid through' sheath scale-arms O and 'IV 'th vbase DA; and a triangle ACB, f

O and T coincide The 'sum ofl these two readings will `N1unti1 CB is equivalent to the length of the otherside of the triangle, and clamping ysc revv R1 holds it in place; mechanism B is slid alongthe'base-bar 4until AB is equiva-y 125 lent tothe iengthof the base of the triangle, and clamp P1 holds it in place. As these operations fare performed, the scale-arm T Awill pivot onr protractor J2, and plate L1'Will turn in pro-- tractor J1; We now have the triangle with Z, therplate L2 willy turn in" anims' the three given sides; and weI can read the angle ACB-on Vernier X1, the angle ABC lon Vernier M1, andangle CAB on Vernier M2'. The mechanism D'is now slid on` the base-bar until Vernier M reads 90 de s, when the arm O will be at right ang es'to the base bar'E'-F, and to DB. The reading CD on Vernier S is then the equivalent of the altitude; and the area of the triangle CAB equals ABxCD/Q.

If the trian le CAB has three acute an les the altitude D will intersect the base B. In constructin the triangle CAB the mechanism D .will side along the base-bar past the mechanism A until it lies at a point on the base-bar between A andB.

In constructing similar ri ht triangles weA can proceed in two ways. f the Jscale-arm U is used as the common side, it is swung until the zero of Vernier M,- reads on 90 derees of the protractor J 2. Clamping screw is then locked. The arm U is then slid through sheath N2 until CA is equivalent to the length of the side common to the two triangles. If the bases of the .two' similar triangles are given, then the mechanisms D and B are slid on thebase-bar, D to the left and B to the right, untilDA and AB are e uivalent to the lengths of the two bases. Cl) and CB will then be equivalent tothe n lengths of the two hypotenuses; and we can read the several acute angles on verniers X,

' X1, M and M1. yIf the scale-arm O is used as the common side, thel mechanism`D is slid to thev right of mechanism A; and AD and DB are made equivalent to the lengths of the two bases. Scale-arm O is then swung until Vernier M reads 90 degrees on protractor J; and arm O is slid through sheath N until CD is equal to the equivalent ofthe length of the common side, the length being read on Vernier S.' The several acute angles may then be read on verniers X, X1, M2. and M1, and the lengths of the two hypotenuses -on scale-arms U and T.

lf the angles of similar triangles are given, we mayselect one of the scale-arms,

as U, for the common side, and set the vernier M2 to read the angles enclosed by the common side, and the bases. We then set mechanism D away from mechanism A a distance so that DA will equal the length of one of the basesA to 'any assumed scale.

Scale-arm U is now slid through sheath N2 'y until angle CDA equals the angle tobe enclosed by base DA and side DC. Similarly,

mechanism B is slid away romA until the angle CBA is equal to the angle to be enclosed by base AB and the side BC. The angles DCA and BCA may thenbe'read on verniers X and X1; and the lengths of the sides DC, and BC, and AC, may be read on f scale-arms O, T and U respectively, and the ylengths of the two bases on scale H, to the scale selected when `assuming the length of ven, may

and formin given angle CAB with the base line a-b. win arm T on pivot Z to form given angle A B; read on Vernier X1. e lamp screw T2. As arm T is swung on Z, mechanism'B will slide to its place on base- .bar EF, formingbase AB and the remaining given angle A C. Now swing arm O on ivot Z, at the same time sliding mechanism on base-bar E-F, until the sum of 'the readings of Vernier X1 (which is now fixed by clamp T2) and Vernier X equals the reading of angle CAB on Vernier M2. Angle C B will then be equal to angle ACB, which is true because angle DBC is common to the similar triangles (JAB and DCB. j

If it be required to find the height of and the distance to an inaccessible object, the summit of which we will designate as S3; loosen all clamping screws on the instrument', and place the bar E-F in a horizontal position (which may be checked with a. bub le), with the scale-arms yO, U, T in a vertical lane; Sight on S3 over the edge of arm A by means of clamping screw Q2. Move a convenient distance back to a second point of observation, and slide mechanism B along the base-bar E-F so that AB will be equivalent to the distance between the two points 'of observation, and secure B by clamping screw P1. Slide U through sheath N2; and at the same time slide arm T in sheath N1, and also rotate T about B, until S, is sighted along the edge of arm T. Set clamping screw Q1. This fixes the triangle ABC. Now slide mechanism D along the base-bar E-F, allowing arm O to pivot on Z and alsov to slide in sheath N, until arm O is perpendicular to the base-bar E-F, readin on Vernier'M. Then CD is the height of the distant object; and AD and BD are its horizontal distances from the first and second points ofi observation, respectively. For

convenience in sighting, sights of equal iis scopic sights'may be usedin the same way.

From the foregoing descriptions of solutions of triangles on my trinoineter, it will be seen that other problems may be solved. For instance; thezsolution of a triangle, having given two sides and the included angle,

or two angles and the included side; or three sides; or two sides and the angle oppositer one of'them. e

' To illustrate the extraction of a square root, of any number, n, yset arm U at right angles to (iL-b; slide mechanism B until AB AD is equal to unity, to the same scale at n.

With these two lengths now fixed, slide arm U through sheath N2 until the angleDCB is 90 degrees. Then the' length ACis the desired squareroot; for,by similar triangles, AC2=AB AD=a 1; and therefore AC=1/7`1 To square a number n, the arm U ,is adjusted to make AC equalftoy a and AD is made equal to unity to the `same scale; then swing T on pivot Z until DCB is 90 degrees. Then AB is equal to n2; for, by

similar triangles, 1 `AB=AC2, "or AB=-n?. The natural trigonometrical\functions of all acute anglesmay also be found on-the rtrinometer. To illustrate: Swing arm U to the 90 degree position, screw Q2. units by sliding arm'T in sheath N1, and set clamping screw R1. Angle CAB `and side BC are now fixed. Now form different valuesof angle ABC by sliding arm U in sheath N2, vat the same time sliding mechanism B along the base-bar. For each value df angle ABC, AC and AB Will give' the values of the corresponding sine and'- cosine, respectively. To obtain tangents and secants; fix AB equal to any tens of units. and set U at right angles to AB. Obtain differ-` ent .angles ABG by sliding arm U in sheath N 2, and by sliding arm T in sheath Ni, at the same time'rotating arm T on pivot Z and and set clamping about B. For each value of angle ABC, AC y will give the tangent, and BC the secant of angle ABC. To obtain cotangents and cosecants; tix AQ equal to any tens of units. Slide mechanism B on thebase-bar to form angles ABC, at the same time allowing arm T to slide through sheath N1. Then for each value of Jangle ABC, AB is the cotangent, and BC is the cosecant, of the angle ABC. Y

In the description of some of the uses of my trinometer I have generally used the arm O to measure altitude of triangles. lSo also may either of the arms U and T be used to measure altitudes. For instance O is used to measure the altitude of a triangle ABC; U to measure the altitude of -a triangle DBC; and T to measure the altitude of a triangle DAC.

In the drawings the circular scales are shown graduated in degrees. The graduations between the zeros of the scale W adjacent to arm U may be -numbered in any convenient way or left blank. 0n these graduations the angles enclosed by the arms may be read approximately.v The graduations of the straight scales may be in any preferred units. In those cases when the protractor J2 is made slidable on the rear of the base-bar, a scale, corresponding to the scale H, is engraved on the top of the basebar for convenience in reading the position of the protractor with reference to the zero Make` BC equal to Iany tens of of HQ Or, lif itis preferred',ia scale similarA to H, and corresponding with it, is'en raved on' the back of the .base-ba r."" It will' e unthe rear of the a Vernier simishows the 'three angles of the triangle; and

the sum of those three angles must be equal to 180 degrees. For this reason my trinometer has great advantages Vover any ,instru` ment in which the third anglel must be found by subtracting the sum of two angles from 180 degrees. y' 1 A triangle, or any bounded by lines; not y members having breadth andthickness. The parts of an instrument4 must have breadth and thickness, as Well as length; and the vinstrument such as mytrmometer is constructed so that its measuring parts are in fact lines'lying upon one face or another of the several members of the instrument. For instance: in Figure 1 of the drawingsy the arms O, U, and T have length, breadth,and thickness; but for an understanding of the use oftheinstrument they may be assumed to be graduated straight lines pivoted together at a common' point-.1' (The graduated edges of the arms, produced, will pass through the pivot). or structural rea-sons the base-bar has breadth and thickness; and t-he protractors attached to the base-bar are mounted with their centers somewhat above it; pally to give a clear sight of the verniers which register the positions of the protractors on the base-bar. But for an understanding of the use of the instrument, the baseebar may be considered to pass through the centers of the protractors, as the line a-b; and the points of intersection of the arms or lines O, U, and T with the base of the triangle under investigation, are the points of intersection of the said lines O, U, and T with the line 1 -b; which line passes through the axes of the protractors. So, geometrically speaking, the lines O, U, and T pass through the center of pivot Z; and they also pass and slide through, and turn about, the centers of the protractors J, J1, and J2, which centers lie on, and are adjustable Wit-h reference to one another on, the line a-b; which line is the base of the triangle or triangles under investigation when the lines O, U, and Tare the` sides. The protractors are shown annular in form. The shape is immaterial, for a protractor is an instrument graduated for the purpose of measuring angles; and it may be vof any external shape so long as it offers a plane surface` on which angular graduations may be marked. The center of the protractor is tgeometric figure, is

'princi- `dei-stood that when protractor J2 (or mechi the point from which the arligular graduabase of the triangle or triangles under investigation. The protractor Vv shown in Figure 1 I termthe fioating protractor, as

' it may be moved, with the arm pivot, to the right or left of the position shown, or nearer to or further away from the base-bar, while remaining in the plane of the instrument. In graduating the arms O, U and T I measure from zero at the center of -the pivot Z. In graduating the base-bar I prefer toplace the zero at a point midway of the base ar; or where rotractor J2 is fixed, at a point projects erpendicularly .by line o-d onto the basear from the centervof J ,L The z eros of verniers K and K1 are pro- Ajected perpendicularly to the base-bar from the centers of protractors J and J1.

Inthe description of my instrument I.

have used the capital letters A, B, C, and D to designate the points of intersection of lines forming the triangle or triangles under investigation; as for instance the point A, the point of intersection between line U and line a-b; the point B, the point of intersection bet-Ween line T and Athe line -b; the

" point C, the point of intersection between the three lines O, U, and T; the point D, the point of intersection between the line O `and the line a--b. I have, so designated `these points because it is customary to so s designate similar points in geometry. For

the sake of simpllfying the description I have also designated the assembled protractors and their parts as mechanisms For.-

instanc'e: the assembled protractor J1 at the point B is designated as the mechanism B; the assembled protractor J2 and its parts located at the point A is designated as the mechanism A. This method of designating the points of the triangles, and the mechanism assembled at these points, makes the descriptiony ofthe instrument shorter and more easily understandable by the reader.

I claim: v

1. In an instrument of the class described: y

tion of the pivotedv arms 'with the added arm may be varied; and means for measurlng the angular relations of the pivoted arms to the added arm.

2. In an instrument of the class described: a graduated base-bar; a plurality of sliding members adjustably arran d on the basebar; sheaths pivoted on t e sliding members and so arranged that the pivots of the sheaths are in one plane; a. plurality of graduated arms pivoted together at one end, and arranged to slide in the sheaths; means for measuringthe angular relations of the arms to one another; and means for measuring the angular relations of the arms to the plane of t e sheath pivots.

3. In an instrument of the class described: a graduated base-bar;- a plurality of protractors slidably mounted in line on the base- 'i lane passing through the centers of `the ase-bar protractors.

4. In an instrument of the class described: a graduated base-bar; two protractors mounted on the base-bar, and adjustable as to their distance from one another; pivots arranged on the centers of the two protrac-r tors; two graduated arms pivoted together at one end and slidably mounted on the protractor pivots so that the lengths of the arms intercepted between the arm pivot and the protractor pivots may be varied, and so that the angular relations between the arms and the plane of the pivots may be varied; means for measuring the angular relations between the arms another protractor mounted on thel base-bar, its center'in line with the centers of the other protractors; a pivot arranged on the center of the third protractor;` a graduated arm pivoted at a common point with vthe first two arms and slidably mounted on the pivot of the third protractor so that the length of the third arm intercepted between the arm pivot and the pivot of the third protractor may be varied; means for varying the angular relation of the third arm to the plane of pivots ofthe protractors;

and means for measuring the angular relationbetween the third arm and the first twoal'lllS.

5. In an instrument of the class described:

a base-bar; a protractor member' fixed on the base-bar; two similar protractors slidably` arran ed on the base-bar and xiposed one on eit er side of the fixed pivo protractor; three graduated'arms pivoted together -i'f-'at'one send;

l other armand with the base-bar.

Ilocking the and arranged -to slide on and turn about the centers of the protractors on the base-bar; and means for fixing the angular relations of any one of the arms with any 6. In an instrument of the class described: the instrument as described in claim 5; and means for fixing thelengths of arm or of bae-bar intercepted between the ivots.

I a base-bar; a protractor fixed on the basebar; two similar protractors slidably mounted on the base-bar, and disposed one on either side of the fixed protractor; sheaths pivotally mounted on the centers of the protractors; means lfor fixing the angular relations of the sheaths to the protractors; three arms pivoted together at one end, and adapted to engage in the sheaths; means for three arms in their angular relation to one arms being such that the lengths ofthe sides of the triangles formed by the arms and the base-bar may be varied without changing the angular relations of the arms to one another and to the base-bar.

`8. In an instrument of the class described :4

a graduatedbase-bar having two scales extending along its length. in opposite direcscales coinciding on one line; a protractor mounted onthe basebar and having its center registering with the zero of the base-bar scales; two portractors mounted slidably on the base-bar so that they may pass the first protractor, and so that a line passing through he centers of the three protractors lies ina plane parallel to the base-bar; means for fixing each of the sliding protractors in positionulndependently; sheaths pivotally mounted on the cenn an instrument of the class escribed .to one anot another; the arrangement of the' of each sheath; a graduated arm slidablyv mounted in the'sheath on the first protractor; means for fixing the arm in the sheath; two graduated arms pivoted to the first arm at one' end, andslidably mounted in the sheaths on the slidably mounted protractors; means for fixing the arms in the sheaths on the slidably mounted protractors; means for measuring to each of the other and means for fixing the angularity of the three arma er; means for fixing the angular of the ythree arms to- 'l graduated arms pivoted together at one end and slidably mounted, one inv the 'sheath on one ofthe base-bar protractors, and the other in the sheath on the floating protractor;v a third graduated arm pivoted onthe center of the floating f protractor and mounted in the sheath on the lother protractor; the three arms and the base-bar forming a quadrilateral figure; means for measuring the lengths of 'the arm's and' the base-bar intercepted between the pivots; and means for measuring the angles included between the four'sides of the figure.

In testimony whereof I have hereunto affixed my signature.

JOSEPH EUGENE ROWE.

base-bar 

